Abstract
We study the large time asymptotics of solutions to the Cauchy problem for the nonlinear nonlocal Schrödinger equation with critical nonlinearity i∂tu-∂x2u+∂x2u-a∂x4u=λu2u,t>0,x∈R,u0,x=u0x,x∈R,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} \\left\\{ \\begin{array}{l} i\\partial _{t}\\left( u-\\partial _{x}^{2}u\\right) +\\partial _{x}^{2}u-a\\partial _{x}^{4}u=\\lambda \\left| u\\right| ^{2}u,\\text { } t>0,{\\ }x\\in {\\mathbb {R}}\\mathbf {,} \\\\ u\\left( 0,x\\right) =u_{0}\\left( x\\right) ,{\\ }x\\in {\\mathbb {R}}\\mathbf {,} \\end{array} \\right. \\end{aligned}$$\\end{document}where a>frac{1}{5},lambda in {mathbb {R}}. We continue to develop the factorization techniques which was started in papers Hayashi and Naumkin (Z Angew Math Phys 59(6):1002–1028, 2008) for Klein–Gordon, Hayashi and Naumkin (J Math Phys 56(9):093502, 2015) for a fourth-order Schrödinger, Hayashi and Kaikina (Math Methods Appl Sci 40(5):1573–1597, 2017) for a third-order Schrödinger to show the modified scattering of solutions to the equation. The crucial points of our approach presented here are based on the {mathbf {L}}^{2}-boundedness of the pseudodifferential operators.
Highlights
We study the large time asymptotics of solutions to the Cauchy problem for the nonlinear nonlocal Schrodinger equation with critical nonlinearity i∂t u − ∂x2u + ∂x2u − a∂x4u = λ |u|2 u, t > 0, x ∈ R, u (0, x) = u0 (x), x ∈ R, where a
Multiplying Eq (1.1) by the operator 1 − ∂x2 −1, we rewrite it in the pseudodifferential form i∂tu − Λu = λ 1 − ∂x2 −1 |u|2 u, t > 0, x ∈ R, u (0, x) = u0 (x), x ∈ R, (1.2)
As far as we know, there are no results on the large time asymptotics of solutions of the Cauchy problem
Summary
Schrodinger equation introduced by [30] to describe the nonlinear propagation of pulses through optical fibers. It arises in the context of high-speed soliton transmission in long-haul optical communication system [12]. Where the linear pseudodifferential operator Λ = 1 − ∂x2 −1 −∂x2 + a∂x4 is characterized by its symbol. As far as we know, there are no results on the large time asymptotics of solutions of the Cauchy problem (1.1). The difficulty of the small data scattering problem lies in the slow time decay rate of the L∞-norm of solutions to the linear problem. The problem on the large time asymptotic behavior becomes more
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callMore From: Zeitschrift f\xfcr angewandte Mathematik und Physik
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
